Name
Student ID #
Math 1320-004
April 24, 2014
FINAL EXAM
Instructions: Show all of your work for full credit.
Problem
1.
2.
3.
4.
5.
Points
20
20
20
20
20
TOTAL
Score
(20 points) 1. Suppose r(t) = ht + 2, 1, t2 i gives the position of an object at time t.
Find the unit tangent vector to the path, and the tangential component
of acceleration, at time t = 1.
(20 points) 2. An angry bird is launched from a slingshot on the (flat) surface of a
planet, at an angle of 60◦ from horizontal, and leaves the slingshot at
61
m. Acceleration due to gravity is 2 m/s2 . How
1 m/s at a height of 16
far does the bird travel in the horizontal direction?
1
(20 points) 3. Suppose v(x, y) = x2 y + ey , where x(r, s) and y(r, s) are unknown. If
x(1, 2) = 1, xr (1, 2) = −3, xs (1, 2) = 5, y(1, 2) = −4, yr (1, 2) = −1,
∂v
(1, 2).
and ys (1, 2) = 2, find
∂s
(20 points) 4. Find and classify (using the Second Derivatives Test) local maxima,
local minima, or saddle points of the function f (x, y) = x4 + y 4 − 4xy.
2
(20 points) 5. Find the absolute minimum and absolute maximum of f (x, y) = x2 ye−x−y
over the square with vertices (0, 0), (3, 0), (0, 3), and (3, 3).
3